The subject-verb inversion is necessary when the sentence starts with an adverb or when the sentence is interrogative. So, the given sentence does not require subject-verb inversion. Therefore, 'where he could' should replace 'where could he' to rectify the given sentence.
Hence, option (a) is the correct answer.
The correct sentence is: His father does not know where he could have gone at that time.
Note:
Interrogative Sentence: An interrogative sentence is a type of sentence that asks a question, as opposed to sentences that make a statement, deliver a command, or express an exclamation. An interrogative sentence ends with a question mark.
Adverbs: Words that describe a verb, an adjective, another adverb, or even a sentence as a whole are called Adverbs.
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